On solutions of Navier-Stokes equatons in invariant spaces
이재문
27동 325호
0
476
06.12 09:04
| 구분 | HYKE |
|---|---|
| 일정 | 2026-07-08(수) 09:30~11:30 |
| 세미나실 | 27동 325호 |
| 강연자 | Changfeng Gui (University of Macau) |
| 담당교수 | 하승열 |
| 기타 | HYKE 화랑 세미나 |
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일시: 2026년 7월 8일 (수) 10:00 - 10:50
장소: 27동 325
연사: Changfeng Gui (University of Macau)
발표제목: On solutions of Navier-Stokes equatons in invariant spaces
초록: The steady Navier-Stokes equations enjoy a special scaling property thanks to its nonlinear character. Several scaling-invariant classes motivated by the scaling property have proved useful in investigating various properties of a solution. On the other hand, a regularity problem of the steady case in higher dimensions (especially 5D) has attracted the attention of many researchers as it is a steady version of the famous regularity problem for the 3D evolutionary case. One can examine scaling-invariant classes, a borderline case not covered by standard regularity theory, to study special scenarios of a possible singularity. In this talk, we shall present a rigidity result to a most general scaling-invariant class and a regularity result eliminating a more general possibility of singularity for steady Navier-Stokes equations in high dimensions. We also show that the steady incompressible Navier-Stokes equations with any given $(-3)$-homogeneous external force on $\mathbb{R}^n\setminus\{0\}$, $4\leq n\leq 10$, have at least one $(-1)$-homogeneous solution which is self-similar and regular away from the origin. The global uniqueness of the self-similar solution is obtained as long as the external force is small.
We will also present a recent result on the global existence of forward self-similar solutions to the two-dimensional incompressible Navier-Stokes equations for any divergence-free initial velocity that is homogeneous of degree $-1$ and locally H\"older continuous. This result requires no smallness assumption on the initial data. In sharp contrast to the three-dimensional case, where $(-1)$-homogeneous vector fields are locally square-integrable, the major difficulty for the 2D problem is the criticality in the sense that the initial kinetic energy is locally infinite at the origin, and the initial vorticity fails to be locally integrable, so that the classical local energy estimates are not available. Our key ideas are to decompose the solution into a linear part solving the heat equation and a finite-energy perturbation part, and to exploit a kind of inherent cancellation relation between the linear part and the perturbation part. These, together with suitable choices of multipliers, enable us to control the interaction terms and to establish the $H^1$-estimates for the perturbation part.
